We will now see how to determine the equation of these lines. Suppose that $<a_1, a_2, a_3>$ is the unique line that passes through the distinct points $\mathbf{p}, \mathbf{q} \in \mathbb{P}^2(\mathbb{R})$. Then the line $<a_1, a_2, a_3>$ is the set of points $\mathbf{x} = [x_1, x_2, x_3] \in \mathbb{P}^2 (\mathbb{R})$ which satisfy:

(1)

\begin{align} \quad a_1x_1 + a_2x_2 + a_3x_3 = 0 \end{align}

In particular, the points $\mathbf{p}, \mathbf{q}$ satisfy the equation above. Now consider the determinant equation:

Therefore the points $\mathbf{p}, \mathbf{q}$ lie on the line given by the determinant equation above. Since the line that passes through the points $\mathbf{p}$ and $\mathbf{q}$ is unique, we have that: